Question

1 + 2 - 3 - 4 + 5 + 6 - 7 - 8 ...- 2019​

Answer 1

Ответ:

0

Объяснение:

Перепишем в следующем виде и группируем по 4 числа

(-0 + 1 + 2 - 3) +(- 4 + 5 + 6 - 7) +(- 8 + 9 + 10 - 11) +...+(-2016 + 2017 + 2018 - 2019)=0+0+...+0=0

1) Легко увидеть закономерность: числовые выражения каждой скобки начинаются с числа кратного на 4.

2) Для любого n∈Z : -n + (n+1) + (n+2) - (n+3) = -n + n + 1 + n + 2 - n - 3=0

Answer 2

To find the value of the given arithmetic series, we can observe that the numbers alternate between addition and subtraction.

Let's group the numbers and signs to simplify the calculation:

(1 + 2) - (3 + 4) + (5 + 6) - (7 + 8) + ... - (2019)

Within each parentheses, we can evaluate the sum.

(1 + 2) equals 3. (3 + 4) equals 7. (5 + 6) equals 11. (7 + 8) equals 15. ... (2019) equals 2019.

Now let's consider the signs. Since the first number is positive, and the pattern alternates between addition and subtraction, the sign depends on the position.

The first group is positive, the second is negative, the third is positive again, and so on.

Since there are 2019 numbers, we can calculate the number of groups: 2019 / 2 = 1009.5 groups.

The .5 indicates that the last group contains only one number.

Since there are an odd number of groups, the last group will be positive.

Now we can sum up the groups:

3 - 7 + 11 - 15 + ... + 2019

We can see that the pattern repeats every four numbers: +3, -7, +11, -15.

To find the sum, we need to calculate how many times this pattern repeats.

Since there are 1009 complete patterns, we can multiply the sum of the pattern (+3 + (-7) + 11 + (-15)) by 1009:

1009 * (3 - 7 + 11 - 15) = 1009 * (-8) = -8072.

Finally, we need to add the last positive number in the sequence, which is 2019:

-8072 + 2019 = -6053.

Therefore, the value of the given arithmetic series is -6053.